Chapter 15, Problem 1SP

(a)

To determine

The mass per unit length of the rope.

Answer to Problem 1SP

The mass per unit length of the rope is $0.267\text{kg}/\text{m}$.

Explanation of Solution

Given Info: The length of the rope is $12\text{m}$ and mass of the rope is $3.2\text{kg}$.

Write the formula to calculate the mass per unit length of the rope.

$\mu =\frac{m}{L}$

Here,

$\mu$ is the mass per unit length of the rope

m is the mass of the rope

L is the length of the rope

Substitute $3.2\text{kg}$ for m and $12\text{m}$ for L in the above equation to calculate $\mu$.

$\begin{array}{c}\mu =\frac{3.2\text{kg}}{12\text{m}}\\ =0.267\text{kg}/\text{m}\end{array}$

Conclusion:

Therefore, the mass per unit length of the rope is $0.267\text{kg}/\text{m}$.

(b)

To determine

The speed of the wave on the rope.

Answer to Problem 1SP

The speed of the wave on the rope is $13.5\text{m}/\text{s}$.

Explanation of Solution

Given Info: The tension on the rope is $48.6\text{N}$ and mass per unit length is $0.267\text{kg}/\text{m}$.

Write the formula to calculate the speed of the wave on the rope

$v=\sqrt{\frac{F}{\mu }}$

Here,

F is the tension on the rope

v is the speed of the wave

Substitute $48.6\text{N}$ for F and $0.267\text{kg}/\text{m}$ for $\mu$ in the above equation to calculate v.

$\begin{array}{c}v=\sqrt{\frac{48.6\text{N}}{0.267\text{kg}/\text{m}}}\\ =13.5\text{m}/\text{s}\end{array}$

Conclusion:

Therefore, the speed of the wave on the rope is $13.5\text{m}/\text{s}$.

(c)

To determine

The wavelength for the wave on the rope.

Answer to Problem 1SP

The wavelength for the wave on the rope is $3\text{m}$.

Explanation of Solution

Given Info: The speed of the wave is $13.5\text{m}/\text{s}$ and frequency of the wave is $4.5\text{Hz}$.

Write the expression for the speed of the wave.

$v=f\lambda$

Here,

$\lambda$ is the wavelength of the wave

v is the speed of the wave

f is the frequency of the wave

Substitute $13.5\text{m}/\text{s}$ for v and $4.5\text{Hz}$ for f in the above expression to calculate $\lambda$.

$\begin{array}{c}\lambda \left(4.5\text{Hz}\right)=13.5\text{m}/\text{s}\\ \lambda =\frac{13.5\text{m}/\text{s}}{4.5\text{Hz}}\\ =3\text{m}\end{array}$

Conclusion:

Therefore, the longest possible wavelength is $3\text{m}$.

(d)

To determine

The number of cycles of the waves.

Answer to Problem 1SP

The number of cycles of wave on the rope is $4.0\text{cycles}$ .

Explanation of Solution

Given Info: The length of the rope is $12\text{m}$ and wavelength of the rope is $3\text{m}$.

Write the expression to calculate the number of cycles on the rope.

$n=\frac{L}{\lambda }$

Here,

n is the number of cycles on the rope

L is the length of the rope

Substitute $12\text{m}$ for L and $3\text{m}$ for $\lambda$ in the above equation to calculate n.

$\begin{array}{c}n=\frac{12\text{m}}{3\text{m}}\\ =4.0\text{cycles}\end{array}$

Conclusion:

Therefore, the number of cycles of wave on the rope is $4.0\text{cycles}$.

(e)

To determine

The time required to travel the wave from one end to the other end.

Answer to Problem 1SP

The time required is $0.89\text{s}$.

Explanation of Solution

Given Info: The length of the rope is $12\text{m}$ and speed of the wave is $13.5\text{m}/\text{s}$.

Write the expression to calculate the time required for the wave to travel to other end.

$t=\frac{L}{v}$

Here,

t is the time required for the wave for the travelling

Substitute $12\text{m}$ for L and $13.5\text{m}/\text{s}$ for v in the above equation to calculate t.

$\begin{array}{c}t=\frac{12\text{m}}{13.5\text{m}/\text{s}}\\ =0.89\text{s}\end{array}$

Conclusion:

Therefore, The time required is $0.89\text{s}$.